Question

Anja simplified the expression StartFraction 10 x Superscript negative 5 Baseline Over Negative 5 x Superscript 10 Baseline EndFraction to StartFraction 15 Over x Superscript 15 Baseline EndFraction. What mistake did Anja make? She divided the exponents instead of subtracting them. She divided the coefficients instead of subtracting them. She subtracted the coefficients instead of dividing them. She subtracted the exponents instead of dividing them.

Answer 1

The mistake that Anja made during the simplification of the considered expression is given by: Option C:  She subtracted the coefficients instead of dividing them

What are coefficients?

Constants who are in multiplication with variables are called coefficients of those variables.

For example, in $10x^2$ we have 10 as coefficient of $x^2$

Those variables who have got no visible coefficient has 1 as their coefficient. Thus, $x^2$ has got its coefficient as 1. It is true since 1 multiplied with any number is that number itself.( $x^2 = 1 \times x^2$ )

What are some basic properties of exponentiation?

If we have $a^b$ then 'a' is called base and 'b' is called power or exponent and we call it "a is raised to the power b" (this statement might change from text to text slightly).

Exponentiation(the process of raising some number to some power) have some basic rules as:

$a^{-b} = \dfrac{1}{a^b}\\\\a^0 = 1 (a \neq 0)\\\\a^1 = a\\\\(a^b)^c = a^{b \times c}\\\\ a^b \times a^c = a^{b+c} \\\\^n\sqrt{a} = a^{1/n} \\\\(ab)^c = a^c \times b^c$

For the considered situation, the expression that Anja is simplifying is:
$\dfrac{10x^{-5}}{-5x^{10}}$

Anja simplified it to $\dfrac{15}{x^{15}}$

She performed operations with variable 'x' correctly since

$\dfrac{x^{-5}}{x^{10}} = \dfrac{1}{x^5} \times \dfrac{1}{x^{10}} = \dfrac{1}{x^{15}}$

But coefficients will also get divided. Instead of dividing 10 by -5, she subtracted them (as 10 - (-5) = 10 + 5 = 15)

This was her mistake.

The correct simplification would be: $\dfrac{10x^{-5}}{-5x^{10}} = \dfrac{10}{-5} \times \dfrac{x^{-5}}{x^{10}} = -2 \times \dfrac{1}{x^{15}} = \dfrac{-2}{x^{15}} = -2x^{-15}$

Thus, the mistake that Anja made during the simplification of the considered expression is given by: Option C:  She subtracted the coefficients instead of dividing them

Learn more about exponent and bases here:

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